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I need to give an example of two orthogonal vectors that are not linearly independent.

My Workings:

I know that linear dependency of vectors $u$ and $v$ means that there exists number $a$ and $b$, which both cant be 0), such that $au+bv = 0 $. So $0=(au+bv)^T(au+bv) = a^2||u||^2+b||v||^2 $ because they are orthogonal. SO $ A\ne = 0$, then $u=0$. Thus, one of the vectors must be the zero vector.

Thus, would $(1,1)^T$ and $(0,0)^T$ be an example of two orthogonal vectors that are not linearly independent?

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  • $\begingroup$ I see a zero vector? That is never lin. independent with any vector. Normally, I am not aware of two orthogonal nonzero vectors that are linear dependent. $\endgroup$ – imranfat Nov 24 '16 at 3:46
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Yep! The vector 0 is orthogonal to all vectors and linearly dependent with all vectors.

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  • $\begingroup$ is (1,1) and (0,0) are two orthogonal vectors that are not linearly independent? $\endgroup$ – jh123 Nov 24 '16 at 4:02
  • $\begingroup$ Yep. $\{(0,0), (a,b)\}$ is linearly dependent and orthogonal for any $a,b$. $\endgroup$ – mrob Nov 24 '16 at 4:11

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